Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500695 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2226 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,13)(12,14)(15,16)(17,18)(19,29)(20,33)(21,27)(22,24)(23,28)(25,32)(26,34)(30,38)(31,39)(35,42)(36,41)(37,40)(43,46)(44,48)(45,47);(1,3,4,8)(2,5,6,7)(9,11,12,15)(10,13,14,16)(18,24)(19,30)(20,34)(21,23)(25,38)(29,40)(31,45)(32,37)(39,48)(41,42)(43,47)(44,46);(10,14)(13,16)(17,19,22,32)(18,25,24,29)(20,31,34,46)(21,30,23,37)(26,39,33,43)(27,40,28,38)(35,45,36,44)(41,47,42,48);(2,6)(3,8)(10,14)(13,16)(17,20)(18,26)(19,31)(21,36)(22,34)(23,35)(24,33)(25,39)(27,42)(28,41)(29,43)(30,44)(32,46)(37,45)(38,47)(40,48);(17,21,22,23)(18,27,24,28)(19,30,32,37)(20,35,34,36)(25,40,29,38)(26,41,33,42)(31,45,46,44)(39,47,43,48)Polycyclic presentation:
2^9|a^2=fi,b^2=fg,c^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfgi,c^b=ch,d^b=df,e^b=ei,h^b=hi,h^c=hi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7411571);